The representation theory of a class of infinite-dimensional groups which
are inductive limits of inductive systems of linear algebraic groups leads to
a new invariant theory. In this article, we develop a coherent and comprehensive
invariant theory of inductive limits of groups acting on inverse limits of modules,
rings, or algebras. In this context, the Fundamental Theorem of the Invariant
Theory is proved, a notion of basis of the rings of invariants is
introduced, and a generalization of Hilbert's Finiteness Theorem is given.
A generalization of some notions attached to the classical invariant theory such
as Hilbert's Nullstellensatz, the primeness condition of the ideals of
invariants are also discussed. Many examples of invariants of the infinite-dimensional
classical groups are given.